Cyclotomic torsion on Fermat Jacobians (Q707960)

Let \(C\) be the Fermat curve defined by the equation \(X^p+Y^p+Z^p=0\), where \(p\) is a prime \(> 11\). We denote by \(J\) the Jacobian variety of \(C\). Let \(K = \mathbb Q(\varepsilon)\), where \(\varepsilon\) is a primitive \(2p\)-th root of unity. We denote by \(J^{\infty}\) the group of divisor classes of degree 0 on \(C\) supported on the cusps which are the points \(a_j=(0,\varepsilon^{2j+1},1)\), \(b_j = (\varepsilon^{2j+1},0,1)\), \(c_j = (\varepsilon^{2j+1},1,0),\) where \(j\in\{0,1,\ldots,p-1\}\). We consider the automorphisms \(\alpha\), \(\beta\), \(\gamma\) of \(C\) given by \[ \alpha(X,Y,Z) = (\zeta X,Y,Z), \quad \beta(X,Y,Z)= (X,\zeta Y,Z), \quad \gamma(X,Y,Z) = (X,Y,\zeta Z), \] where \(\zeta = \varepsilon^2\), and the endomorphisms \(\pi = \beta-1\), \(\pi'=\alpha-1\), \(\pi'' = \gamma-1\) of \(J\). In this paper it is proved that \(J(K)_{\text{tors}}\) equals either \(J[\pi\pi'\pi'']\) or \(J^{\infty}\).